Problem 84 - Algebra and Trigonometry

Question

The Height of the Graph of an Exponential Function Your mathematics instructor asks you to sketch a graph of the exponential function $$ f(x)=2^{x} $$ for \(x\) between 0 and \(40,\) using a scale of 10 units to one inch. What are the dimensions of the sheet of paper you will need to sketch this graph?

Step-by-Step Solution

Answer

Use a 4-inch by 6-inch sheet, with a logarithmic scale for the y-axis.

Step 1: Determine Range of x

The function needs to be graphed for \(x\) values ranging from 0 to 40. Since the graph spans 40 units on the x-axis and we're using a scale of 10 units per inch, this requires 4 inches along the x-axis.

Step 2: Calculate y Values

As \(x\) increases, the value of \(f(x) = 2^x\) increases exponentially. For example, at \(x = 0\), \(f(x) = 1\); at \(x = 1\), \(f(x) = 2\); and at \(x = 10\), \(f(x) = 1024\). So, \(f(x)\) continues to grow rapidly.

Step 3: Logarithmic Scale for y-axis

Since \(f(x)\) grows exponentially and rapidly becomes quite large, a logarithmic scale for the y-axis will make more sense. This allows us to fit higher values into a reasonably sized graph and helps visualize the exponential growth better.

Step 4: Calculate Paper Dimensions Needed

Using the largest \(x\) value (40), we determined a 4-inch width is needed. Using an exponential or logarithmic scale for \(f(x)\), we need to determine a height that accommodates the largest value \(f(x) = 2^{40}\), which is very large (\( \approx 1.1 \times 10^{12} \)). If a logarithmic scale accommodates this large number within 6 inches on the y-axis, the paper should be at least 4 inches wide and 6 inches tall.

Step 5: Finalize Paper Size Decision

Considering the scale, and comfortably allowing space for plotting and labeling, a feasible paper size is approximately 4 inches by 6 inches, assuming the logarithmic scaling of the y-axis is implemented.

Key Concepts

Graphing Exponential Functions

Exponential functions like \( f(x) = 2^x \) are characterized by their power to grow swiftly as \( x \) increases. These functions are distinct from linear functions as their rate of increase is not constant but rather grows proportionally to their current value.
To visually represent this growth, we create a graph. **Sketching an Exponential Function** When graphing, we plot values of \( x \) against \( f(x) = 2^x \):

• At \( x = 0 \), \( f(x) = 1 \).
• At \( x = 1 \), \( f(x) = 2 \).
• At \( x = 10 \), \( f(x) = 1024 \).

Notice how the values begin small and increase dramatically, showcasing the unique curvature that exponential graphs exhibit. This growth curve is smooth and rises steeply.

Logarithmic Scale

When graphing functions like \( f(x) = 2^x \), which grow rapidly, it can be challenging to fit large values on a standard linear scale. This is where logarithmic scales come into play.
**Benefits of a Logarithmic Scale**:

• It compresses the larger values into a more manageable space.
• It allows us to better visualize data across large ranges of \( y \) values.

In practical terms, a logarithmic scale allows us to represent values like \( 2^{40} \), which is approximately \( 1.1 \times 10^{12} \), within a reasonable height on paper without losing detail in smaller values. It maps each step as a power of a base, making it ideal for displaying exponential growth.

Function Growth

The defining feature of an exponential function is its rapid growth rate. As\( x \) increases, the function \( f(x) = 2^x \) grows exponentially, meaning:

• For each unit increase in \( x \), the output multiplies by a constant factor, in this case, 2.
• This leads to small initial increases, but explosive growth as \( x \) becomes larger.

**Why Exponential Growth is Significant**:
In many real-world situations, phenomena that exhibit exponential growth often surpass our expectations. Examples include population growth, compound interest, and the spread of viral information. Understanding the rapid acceleration associated with exponential functions is crucial for interpreting and predicting real-world trends.

X-Axis Range

Deciding the range for the \( x \)-axis is a vital aspect of graphing exponential functions. This determines how much of the function's behavior you can visualize.
In the scenario we're exploring, the range is from 0 to 40 for \( x \):

• By using a scale of 10 units per inch, we condense these 40 units into a 4-inch space.
• This compression is sufficient to display the function’s entire growth pattern along the \( x \)-axis.

**Importance of Choosing an Effective Range**:
Choosing the right \( x \)-axis range helps graph the most crucial parts of a function. It ensures that the main behavior of the function is visible and interpretable, facilitating accurate analysis.